In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass determining the gravitational force is not reduced. In the computation one mass can be replaced with the reduced mass, if this is compensated by replacing the other mass with the sum of both masses. The reduced mass is frequently denoted by , although the standard gravitational parameter is also denoted by . It has the dimensions of mass, and SI unit kg.
Equation
Given two bodies, one with mass m1 and the other with mass m2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass where the force on this mass is given by the force between the two bodies.
Properties
The reduced mass is always less than or equal to the mass of each body: and has the reciprocal additive property: which by re-arrangement is equivalent to half of the harmonic mean. In the special case that : If, then.
Derivation
The equation can be derived as follows.
Newtonian mechanics
Using Newton's second law, the force exerted by a body on another body is: The force exerted by particle 1 on particle 2 is: According to Newton's third law, the force that particle 2 exerts on particle 1 is equal and opposite to the force that particle 1 exerts on particle 2: Therefore: The relative acceleration arel between the two bodies is given by: Note that, the relative acceleration is equal to the acceleration of the separation between the two particles. This simplifies the description of the system to one force, one coordinate, and one mass. Thus we have reduced our problem to a single degree of freedom, and we can conclude that particle 1 moves with respect to the position of particle 2 as a single particle of mass equal to the reduced mass,.
Alternatively, a Lagrangian description of the two-body problem gives a Lagrangian of where is the position vector of mass . The potential energyV is a function as it is only dependent on the absolute distance between the particles. If we define and let the centre of mass coincide with our origin in this reference frame, i.e. then Then substituting above gives a new Lagrangian where is the reduced mass. Thus we have reduced the two-body problem to that of one body.
In a system with two point masses and such that they are co-linear, the two distances and to the rotation axis may be found with where being the sum of both distances This holds for a rotation around the center of mass. The moment of inertia around this axis can be then simplified to
Collisions of particles
In a collision with a coefficient of restitutione, the change in kinetic energy can be written as where vrel is the relative velocity of the bodies before collision. For typical applications in nuclear physics, where one particle's mass is much larger than the other the reduced mass can be approximated as the smaller mass of the system. The limit of the reduced mass formula as one mass goes to infinity is the smaller mass, thus this approximation is used to ease calculations, especially when the larger particle's exact mass is not known.
Motion of two massive bodies under their gravitational attraction
In the case of the gravitational potential energy we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because
Consider the electron and proton in the hydrogen atom. They orbit each other about a common centre of mass, a two body problem. To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass and the proton mass becomes the sum of the two masses This idea is used to set up the Schrödinger equation for the hydrogen atom.
Other uses
"Reduced mass" may also refer more generally to an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. A similar expression appears in the transversal vibrations of beams for the elastic moduli. This relationship is determined by the physical properties of the elements as well as the continuity equation linking them.
